PHYSICAL REVIEW X 10, 011040 (2020) Featured in Physics

Accelerating with External Electric and Magnetic Fields

T. Chervy ,1,* P. Knüppel ,1,* H. Abbaspour ,1 M. Lupatini,2 S. Fält ,1,2 W. Wegscheider,2 M. Kroner,1 and A. Imamogluˇ 1 1Institute of Quantum Electroncis, ETH Zürich, CH-8093 Zürich, Switzerland 2Solid State Physics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland

(Received 15 November 2019; accepted 14 January 2020; published 19 February 2020)

It is widely assumed that cannot be manipulated using electric or magnetic fields. Even though hybridization of photons with electronic polarization to form -polaritons has paved the way to a number of groundbreaking experiments in semiconductor microcavities, the neutral bosonic nature of these has severely limited their response to external gauge fields. Here, we demonstrate acceleration by external electric and magnetic fields in the presence of nonperturbative coupling between polaritons and itinerant , leading to formation of new quasiparticles termed -polaritons. We identify the generation of density gradients by the applied fields to be primarily responsible for inducing a gradient in polariton energy, which in turn leads to acceleration along a direction determined by the applied fields. Remarkably, we also observe that different polarization components of the polaritons can be accelerated in opposite directions when the electrons are in ν ¼ 1 integer quantum Hall state.

DOI: 10.1103/PhysRevX.10.011040 Subject Areas: Condensed Physics, Optics, Photonics

Controlling photons with external electric or magnetic the speed of a polariton superfluid, raising new questions fields is an outstanding goal. On the one hand, coupling and possibilities regarding the interplay between the normal photons to artificial gauge fields holds promises for the and condensed fractions of the fluid in the presence of realization of topological and strongly correlated phases electron-exciton interactions. of light [1–4]. On the other hand, effecting forces on While interactions between polaritons and electrons have photons constitutes both a problem of fundamental interest been proposed and analyzed as a mechanism for polariton in electromagnetism and an important step in view of thermalization [22–25], early studies reported the modifi- technological applications [5–8]. One promising avenue cations to polariton resonances in the presence of a Fermi toward this goal is to hybridize photons with material sea [26–28]. These modifications stem from dispersive excitations that are genuinely sensitive to gauge fields [9]. interactions between the polarizable excitonic component In this nonperturbative regime, exciton-polariton states of the polariton and charge-density fluctuations of the are formed, ensuring that the forces acting on the material Fermi sea [29–31]. In a typical setting where are excitations are directly imprinted onto the . hosted by a GaAs quantum well (QW), the existence of a However, the neutral bosonic nature of polaritons has so -singlet trion bound channel leads to the formation of far severely limited their response to gauge fields [10–13]. two optically active branches referred to as attractive and A particularly appealing approach to circumvent this repulsive [32,33]. These polaronic states consist limitation is to leverage on the interaction between excitons of an exciton dressed by collective trion-hole excitations and charges. Indeed, early reports on the drift of trions in of the Fermi sea, where the spin of the photoexcited an electric field [14,15], as well as on the Coulomb drag electron is opposite to that of the dressing Fermi sea effect in bilayer systems [16–19], indicated that it may be electrons [32–34]. Upon decreasing the electron density, possible to manipulate neutral excitations using electric the weight of the repulsive polaron branch, fields in a solid-state setting. Recently, experimental [20] quantifying its excitonic character, increases. In the limit of and theoretical [21] studies reported the electrical control of vanishing electron density, the repulsive (attractive) polaron asymptotically becomes the bare exciton (molecular trion) *These authors contributed equally to this work. state with strong (vanishing) coupling to the cavity mode [32]. The many-body polaronic states are expected to Published by the American Physical Society under the terms of be charge neutral [31,32], suggesting the absence of the Creative Commons Attribution 4.0 International license. coupling to external fields. In a recent theoretical study, Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, however, it was shown that neutral polarons are sensitive to and DOI. the average force on electrons, leading to a finite polaron

2160-3308=20=10(1)=011040(7) 011040-1 Published by the American Physical Society T. CHERVY et al. PHYS. REV. X 10, 011040 (2020) transconductivity in the nonequilibrium limit—an effect wedge in the cavity length introduced during sample that should be observable even when polarons hybridize growth. The sample is cooled to a base temperature of with cavity photons [35]. 20 mK in a dilution refrigerator with free space optical In the present work, we demonstrate experimentally that access enabling simultaneous position and momentum- external electric and magnetic fields effect forces on polaron- resolved microscopy, as well as magnetotransport experi- polaritons. In contrast to earlier proposals, we find that the ments (see Supplemental Material [37]). observed polariton acceleration primarily originates from a We first characterize the electrical properties of our source-drain voltage induced density gradient in the two- sample by a four-point current-voltage (I-V) measurement dimensional electron gas (2DEG) in which polaritons are yielding the characteristic shown in Fig. 1(c).Atlow immersed. Remarkably, we show that the direction of source-drain voltage bias the I-V curve is linear with a the resulting force can be changed by an externally 2DEG sheet resistivity of 210 Ω=sq. Increasing the source- applied magnetic field, since the induced Hall voltage drain bias to ca. 1 V, the I-V curve becomes nonlinear, creates transverse density gradients. Finally, we extend this entering a regime where the externally applied electro- approach to demonstrate spin-selective acceleration of polar- chemical potential leads to a spatially varying chemical itons when the 2DEG is in the integer quantum Hall regime. potential across the 2DEG. At even larger source-drain The sample used in this study is a monolithic distributed bias, the 2DEG is depleted and we recover a linear I-V Bragg reflector optical microcavity with a quality factor characteristic with an increased resistivity of 1500 Ω=sq, Q ¼ 5.5 × 103 grown by molecular-beam epitaxy. The corresponding to electrical conduction in parallel channels structure contains a single optically active GaAs QW of (see Supplemental Material [37]). 20 nm thickness located at the central antinode of the cavity These different regimes of the 2DEG behavior have field. This QW is remotely doped from both sides by Si a counterpart on the optical response of the system. impurities embedded in thin GaAs QWs (10 nm thickness) Figure 2(a) shows the angular dispersion of the polaron- located at the nodes of the cavity field. The doping QWs polariton states at zero source-drain bias. There, we identify thus provide electrons to the optically active QW where a four branches resulting from the hybridization of the 2DEG is formed with nominal electron density of ne ¼ cavity photon with the attractive and repulsive heavy-hole 0.33 × 1011 cm−2 and mobility μ ¼ 1.6 × 106 cm2 V−1 s−1 polarons and with the light-hole resonance. The dispersive (see Ref. [36] for further details on the sample fabrication). polaron-polariton branches are well reproduced by the To study the interplay between polariton transport and 2DEG physics, the sample is etched in the form of a Hall bar with annealed electrical contacts to the 2DEG, as depicted in Figs. 1(a) and 1(b). A photoluminescence (PL) scan map of the sample, recorded at normal incidence (in-plane momentum kk ¼ 0), is shown in Fig. 1(d) out- lining the shape of the Hall bar. The increase in emission energy of about 3 meV=mm from left to right is due to a

FIG. 2. Electrically controlled polariton landscape. (a) Normalized white-light reflectivity spectra showing the FIG. 1. Electrical and optical properties of the sample. polaron-polariton dispersion at the nominal electron density. (a) Layout of the Hall bar and the contacts. (b) Side view of (b) Exciton-polariton dispersion in the depleted regime the Hall bar. The top mirror has been etched to contact the 2DEG (ΔVL ¼ −6 V). Dashed lines are coupled oscillators fits to the located in the center QW [doping quantum wells (DQW)]. data. (c),(d) Normalized white-light reflectivity spectra at kk ¼ −1 (c) Current-voltage characteristic of the device where VR acts 1.2 μm [vertical line in (a) and (b)] as a function of position with a as current source and VL as drain. (d) Map of the lower polariton negative bias voltage applied to the left contact ðΔVL ¼ −2.4 VÞ PL wavelength measured at normal incidence (kk ¼ 0). (c) and to the right contact ðΔVR ¼ −2.4 VÞ (d).

011040-2 ACCELERATING POLARITONS WITH EXTERNAL ELECTRIC … PHYS. REV. X 10, 011040 (2020) coupled oscillators fit shown in dashed lines. Figure 2(b) shows the angular dispersion acquired while applying a large bias of −6 V to the left contact and putting the right contact to ground, ΔVL ¼ −6 V [see Fig. 1(a)]. There, we measure the typical exciton-polariton dispersion of an intrinsic QW with anticrossings about the heavy-hole and light-hole exciton energies, as expected from the 2DEG depletion observed in the I-V characteristic. In between these two extremes, at intermediate bias voltages, the electron density shows smooth spatial gra- dients across the Hall bar as demonstrated in Fig. 2(c) for ΔVL ¼ −2.4 V. Here, the polariton spectrum is recorded at positions across the long axis of the Hall bar and at fixed −1 collection angle corresponding to kk ¼ 1.2 μm . On the left-hand side, where the negative bias is applied, the sample is devoid of electrons and the spectrum resembles a vertical cut in the dispersion of Fig. 2(b). Moving to the right, the electron density increases and we gradually recover polaron-polariton spectra corresponding to the dispersion of Fig. 2(a) as the oscillator strength is gradually transferred from the excitonic to the polaronic resonances. Reversing the applied electric field (ΔVR ¼ −2.4 V) inverts this density gradient, as shown in Fig. 2(d). Such electron density gradients constitute electrically tunable potential landscapes for polaritons which we explore in the FIG. 3. Polariton acceleration in external electric and magnetic following to transport neutral optical excitations. fields. (a) Normalized difference between two RF images of To demonstrate polariton transport, we resonantly polariton flow in opposing electron density gradients. (b) Com- excite a polariton cloud and image its expansion in two parison with a trajectory-based model (dashed green line). The red and blue line is a line cut of (a) at y ¼ 25 μm and the black line is opposing electron density gradients. The excitation laser, at an average between y ¼ 10 and y ¼ 40 μm. (c)–(f) Normalized 1524.0 meV ð813.54 nmÞ, is linearly polarized and focused I I x ¼ 450 μ difference between images R, L as in (a) for magnetic fields: in the central region of the Hall bar ( m in Figs. 1 (c) B ¼ 8 mT, (d) B ¼ −8 mT (e), B ¼ 40 mT, and (f) B ¼ and 2), thus injecting a radially expanding cloud of polar- −40 mT. The horizontal dash-dotted lines in (a),(c)–(f) delimit the itons with finite kk. The decaying polariton signal is width of the Hall bar, the large circle is the field of view of the collected by the same microscope objective lens and microscope, and the central gray disk is the excitation spot. separated from the scattered excitation beam by polariza- tion filtering. A finite strain along the crystalline axes in the structure allows us to obtain this resonance fluorescence characterize the lower polariton energy landscape E ðx; k Þ (RF) signal by polarizing the excitation beam at 45° with LP x , separately for the two density gradients used respect to the polarization eigenbasis defined by strain to measure IR and IL (see Supplemental Material [37]). (see Supplemental Material [37]). Two images, IR and IL, Starting from the classical Hamilton equations of motion in of this RF signal are acquired under source-drain biases one dimension (x), we propagate the experimentally −1 of ΔVR ¼ −2.4 V and ΔVL ¼ −2.4 V, respectively. These determined initial conditions kxð0Þ¼1.2 μm to pre- two voltages were chosen such that at the injection spot on dict the emission intensity IðxÞ, again repeated for the two the Hall bar, the electron densities are the same and the energy landscapes corresponding to the two opposing gradients are of opposite signs. This choice is not necessary gradients. As in the experiment, we calculate the normal- but simplifies the observation of polariton acceleration, as ized difference between the two cases and compare the there is no trivial difference between dispersions and result (green dashed line) with the experiment in Fig. 3(b). thereby group velocities between the two images to be The red and blue colored line is a line cut through Fig. 3(a) compared. Figure 3(a) shows the normalized difference of at y ¼ 25 μm and the black curve is an average over y the two images, ðIR − ILÞ=ðIR þ ILÞ, clearly demonstrat- between 10 and 40 μm. Remarkably, this simple approach ing the ability to route polaritons by electrical means. allows us to obtain reasonable quantitative agreement with In order to model this effect, we measure the polariton our data. dispersion at the two sides of the field of view [x ¼ 0 and We can further control the polariton flow by applying a x ¼ 50 μm in Fig. 3(a)] and perform a coupled oscillator magnetic field perpendicular to the QW plane. In con- fit. By linearly interpolating across the field of view, we junction with a finite source-drain bias, this induces a Hall

011040-3 T. CHERVY et al. PHYS. REV. X 10, 011040 (2020) voltage transverse to the applied potential leading to charge strength have a direct counterpart in the Rabi splittings redistribution in the y direction [42,43]. The combined Ωσ of the corresponding polaron-polariton branches, electric and magnetic fields now shape the electron density as shown in Fig. 4(a) and investigated in detail in gradient which can be tuned in angle and magnitude, as Refs. [36,44]. The degree of electron spin polarization 2 demonstrated in Figs. 3(c)–3(f). Figure 3(c) corresponds can be inferred from the Rabi splittings as Sz ≃ ðΩσþ − 2 2 2 ΔVR ¼ −2 4 ΔVL ¼ − − to alternating voltage biases of . V and Ωσ Þ=ðΩσþ þ Ωσ Þ ≃ 70% at ν ¼ 1 (1.26 T). In this −2.4 V and a fixed magnetic field of 8 mT and shows the possibility to orient the polariton flow in a diagonal direction. Further increasing the magnetic field to 40 mT [Fig. 3(e)] leads to polariton transport in the up-down direction with an intensity contrast of ca. 50%. Moreover, flipping the sign of the applied magnetic field reverses the direction of polariton transport, as shown in Figs. 3(d) and 3(f). While these results demonstrate the possibility to trans- port dressed photons (i.e., polaritons) by electric and magnetic fields, it should be noted that we do not observe here a Lorentz force for photons. In particular, the force acting on polaritons does not appear to depend on the direction of their motion. As clearly shown in Figs. 3(e) and 3(f), both the polaritons propagating to the left and to the right are deflected in the same direction. The polariton acceleration is determined only by the electron density gradient which in turn is controlled by the combination of magnetic field and electrical bias. In other words, the nonperturbative coupling of polaritons to itinerant electrons allows for the control of photons by electromagnetic forces acting on the electronic sector. In the last part of this article, we demonstrate how this idea can be extended to realize transverse polariton spin currents reminiscent of an intrin- sic spin-Hall effect when the 2DEG is close to the ν ¼ 1 integer quantum Hall state. At low temperature and under a strong magnetic field, the optical excitation spectrum of a high-mobility 2DEG exhibits energy gaps due to the quantization of cyclotron orbits. The corresponding Landau levels are further split in energy by a Zeeman field, forming a ladder of spin subbands for the electrons. As the occupancy of this ladder FIG. 4. Spin-selective polariton acceleration. (a) Normalized k ¼ 1 2 μ −1 is varied (e.g., by tuning the magnetic field), the 2DEG white-light reflectivity spectra measured at k . m as a undergoes phase transitions between electronic ground function of magnetic field. The corresponding Landau level filling factor is determined from an independent magnetotran- states of different spin polarization Sz. In particular, when sport measurement. (b) Normalized reflectivity spectra at kk ¼ the lower spin subband of the first Landau level is 1.2 μm−1 and B ¼ 1.1 T across the vertical y direction on the ν ¼ 1 completely filled ( ), the electronic ground state is Hall bar, with a source-drain bias ΔVR ¼ −0.17 V. Black points an itinerant ferromagnet with strong spin polarization. indicate the electron spin polarization Sz measured at three Remarkably, the fact that polaron dressing occurs exclu- different positions. (c),(d) Momentum-resolved polariton RF sively in the configuration where the optically excited emission under cross-linear polarization at 1523.8 meV electron and the electrons in the 2DEG dressing cloud have ð813.62 nmÞ for (c) ΔVR ¼ −0.17 V and (d) ΔVL ¼ −0.35 V. opposite spin renders the polaron very sensitive to the spin The red dashed lines are guides to the eye. (e),(f) Normalized polarization of the 2DEG. At ν ¼ 1, the polaron oscillator difference between two RF images of right-propagating polar- itons acquired with ΔVR ¼ −0.17 V and ΔVL ¼ −0.35 V. strength is maximum for right-hand circularly polarized − −1 þ (e) Excitation of σ polaritons at kx ¼ −0.9 μm , ky ¼ 0. light σ since most of the 2DEG electrons are in the spin- þ −1 (f) Excitation of σ polaritons at kx ¼ −1.3 μm , ky ¼ 0. up state, thus allowing for efficient dressing. Conversely, The horizontal dash-dotted lines delimit the width of the Hall the oscillator strength is reduced for left-hand circularly bar, the large circle is the field of view of the microscope. The red − polarized light σ due to the absence of spin-down and blue rings in (c)-(f) show the excitation spots in real and electrons. The ensuing variations of polaron oscillator momentum spaces for σþ and σ− polaritons, respectively.

011040-4 ACCELERATING POLARITONS WITH EXTERNAL ELECTRIC … PHYS. REV. X 10, 011040 (2020) quantum Hall regime, the charge-density gradients dem- Rashba-type coupling at the origin of standard spin-Hall onstrated above translate into gradients of 2DEG spin effects. Instead, the interaction reported here is analogous polarization, resulting in optical spin-contrasted forces to a force for photons, where the spatially varying for polaritons. electron spin polarization (Sz) acts as an accelerating pol Figure 4(b) shows the evolution of the polariton spec- potential, sorting polaritons of different spin (Sz )in −1 trum at kk ¼ 1.2 μm , recorded from the lower edge to the different directions: upper edge of the Hall bar (y direction) under a voltage bias of ΔVR ¼ −0.17 V and a magnetic field of 1.1 T. We pol F ∼ Sz ∇Sz: ð1Þ observe four energy branches, as expected from a vertical photon cut in Fig. 4(a) near ν ¼ 1, where the two inner branches are the σ−-polarized lower and upper polaritons and the two It should be noted that the evolution of electron spin outer ones are the σþ-polarized lower and upper polaritons. polarization around ν ¼ 1 quantum Hall plateau is widely In order to calculate the degree of spin polarization, we believed to involve the proliferation of in the acquired energy dispersions at three different y positions quantum Hall ferromagnetic state due to the interplay and extracted the Rabi splittings by a fit based on coupled between Zeeman and Coulomb energies [53]. The spin- oscillators (see Ref. [37]). As can be seen in Fig. 4(b), the singlet polaron-polariton dressing thus constitutes a new degree of electron spin polarization of the 2DEG (black interface for coupling the optical spin of photons to the marks, right axis) evolves across the Hall bar and can be electronic spin excitations of 2DEGs. The behavior of controlled electrically as demonstrated in the Supplemental such interactions in the fractional quantum Hall regime Material [37]. Remarkably, such variations in Sz constitute where excitons may be dressed by fractionally charged gradients of opposite signs for the σþ and σ− polaron- quasiparticles as well as the residual interactions between polariton energy landscapes. these fractional quantum Hall polaritons remain to be To investigate the resulting spin-dependent polariton explored [54]. acceleration, we first perform momentum-resolved measure- In summary, we demonstrated novel ways of controlling ments by imaging the polariton RF emission (see photons with external electric and magnetic fields, which Supplemental Material [37]). An excitation energy of are enabled by their hybridization with polarization waves 1523.8 meV ð813.62 nmÞ is chosen to intercept both σþ in a medium. While these results were obtained in the − and σ lower polariton dispersions at finite kk. Figures 4(c) context of exciton-polaritons interacting with electrons, we and 4(d) show the momentum-resolved polariton RF signal highlight that the underlying mechanism is general and could allow for the electrical control of photons hybridized for ΔVR ¼ −0.17 VandΔVL ¼ −0.35 V, respectively. As can be seen in Fig. 4(c), the inner σ−-polarized branch is with other kinds of polarization waves such as or þ , provided that the quanta of polarization con- shifted toward positive ky while the outer σ -polarized currently interact with free electrons or holes. branch is shifted toward negative ky.Thisobservation A nonequilibrium electron density gradient is shown to directly demonstrates an in-plane acceleration whose sign act as an effective electric field for polaron-polaritons Spol depends on the spin of the polaritons z in the given which is tunable in strength and direction. We foresee that quantum state [45]. The effect is reversed in Fig. 4(d),where this effective electric field could be further controlled by the external bias and thereby the gradient in electron spin tailoring the 2DEG density, e.g., using patterned electrodes. y polarization points in the opposite direction. The real-space By mapping the energy landscape of the lower polariton, counterpart of this acceleration allows for the generation of we reach quantitative agreement between a simple trajec- transverse optical spin currents, reminiscent of an intrinsic tory-based model and the observed polariton acceleration. – optical spin-Hall effect [46 52]. Our experiment constitutes an alternative to the already To demonstrate the generation of transverse polariton proposed polariton drag effect for exerting electromagnetic spin currents, we inject polariton waves of well-defined forces on neutral optical excitations [18,35]. We emphasize momenta by focusing the excitation beam in the back-focal that the electron density gradients we exploit are generic for plane of the objective lens. Figures 4(e) and 4(f) correspond low-density 2DEGs when large source-drain voltages are −1 −1 to excitation at kx ¼ −0.9 μm , ky ¼ 0 and kx ¼ −1.3 μm , applied and therefore need to be considered in view of − ky ¼ 0, resulting in right propagating polaritons with σ polariton drag experiments. and σþ polarization, respectively. The normalized differ- In the integer quantum Hall regime, we demonstrated ence of the propagation images, acquired with the two that electron spin depolarization, induced by the prolifer- different voltage biases, indeed reveals opposite acceler- ation of skyrmions as the electron density gradient pushes ation of σ− and σþ polaritons along the y direction. the system away from ν ¼ 1 filling, constitutes a scalar The spin-dependent momentum shifts demonstrated potential for the optical spin of photons and results in spin- here, although capable of generating transverse spin cur- dependent accelerations. This observation opens a new rents, remain fundamentally different from the usual interface between optical and electronic spin sectors as

011040-5 T. CHERVY et al. PHYS. REV. X 10, 011040 (2020) magnetic excitations of the 2DEG, such as , could Exciton-Polariton Topological Insulator, Nature (London) be used for spin-selective drag of photons. 562, 552 (2018). [14] D. Sanvitto, F. Pulizzi, A. J. Shields, P. C. M. Christianen, The data that support the findings of this article are S. N. Holmes, M. Y. Simmons, D. A. Ritchie, J. C. Maan, available in the ETH Research Collection [55]. and M. Pepper, Observation of Charge Transport by Negatively Charged Excitons, Science 294, 837 (2001). ACKNOWLEDGMENTS [15] F. Pulizzi, D. Sanvitto, P. C. M. Christianen, A. J. Shields, S. N. Holmes, M. Y. Simmons, D. A. Ritchie, M. Pepper, and We thank H.-T. Lim, S. Ravets, E. Togan, and Y. J. C. Maan, Optical Imaging of Trion Diffusion and Drift in Tsuchimoto for extensive help with the design and fab- GaAs Quantum Wells, Phys. Rev. B 68, 205304 (2003). rication of the sample. We also acknowledge fruitful [16] D. V. Kulakovskii and Y. E. Lozovik, Screening and Rear- discussions with O. Cotlet, E. Demler, T. Ihn, P. Märki, rangement of Excitonic States in Double Layer Systems, A. Popert, R. Schmidt, Y. Shimazaki, and T. Smolenski. J. Exp. Theor. Phys. 98, 1205 (2004). This work was supported by the Swiss National Science [17] O. L. Berman, R. Y. Kezerashvili, and Y. E. Lozovik, Can We Move Photons?, Phys. Lett. A 374, 3681 (2010). Foundation (NCCR Quantum Science and Technology). [18] O. L. Berman, R. Y. Kezerashvili, and Y. E. Lozovik, Drag This project has received funding from the European Effects in a System of Electrons and Microcavity Polaritons, Research Council under the Grant Agreement Phys. Rev. B 82, 125307 (2010). No. 671000 (POLTDES). [19] B. N. Narozhny and A. Levchenko, Coulomb Drag, Rev. Mod. Phys. 88, 025003 (2016). [20] D. M. Myers, B. Ozden, J. Beaumariage, L. N. Pfeiffer, K. West, and D. W. Snoke, Pushing Photons with Electrons: Observation of the Polariton Drag Effect, arXiv:1808 [1] L. Lu, J. D. Joannopoulos, and M. Soljačić, Topological .07866. Photonics, Nat. Photonics 8, 821 (2014). [21] I. Y. Chestnov, Y. G. Rubo, and A. V. Kavokin, Pseudo- [2] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Drag of a Polariton Superfluid, Phys. Rev. B 100, 085302 Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, (2019). and I. Carusotto, Topological Photonics, Rev. Mod. Phys. [22] G. Malpuech, A. Kavokin, A. Di Carlo, and J. J. Baumberg, 91, 015006 (2019). Polariton Lasing by Exciton-Electron Scattering in Semi- [3] M. Aidelsburger, S. Nascimbene, and N. Goldman, Artifi- cial Gauge Fields in Materials and Engineered Systems, conductor Microcavities, Phys. Rev. B 65, 153310 (2002). C. R. Phys. Quantum Simul. 19, 394 (2018). [23] P. G. Lagoudakis, M. D. Martin, J. J. Baumberg, A. Qarry, [4] I. Carusotto and C. Ciuti, Quantum Fluids of Light, Rev. E. Cohen, and L. N. Pfeiffer, Electron-Polariton Scattering Mod. Phys. 85, 299 (2013). in Semiconductor Microcavities, Phys. Rev. Lett. 90, [5] J. L. O’Brien, A. Furusawa, and J. Vučković, Photonic 206401 (2003). Quantum Technologies, Nat. Photonics 3, 687 (2009). [24] A. Qarry, G. Ramon, R. Rapaport, E. Cohen, A. Ron, A. [6] D. Sanvitto and S. K´ena-Cohen, The Road towards Polari- Mann, E. Linder, and L. N. Pfeiffer, Nonlinear Emission due tonic Devices, Nat. Mater. 15, 1061 (2016). to Electron-Polariton Scattering in a Semiconductor Micro- [7] C. Schneider, K. Winkler, M. D. Fraser, M. Kamp, Y. cavity, Phys. Rev. B 67, 115320 (2003). Yamamoto, E. A. Ostrovskaya, and S. Höfling, Exciton- [25] M. Perrin, P. Senellart, A. Lemaître, and J. Bloch, Polariton Polariton Trapping and Potential Landscape Engineering, Relaxation in Semiconductor Microcavities: Efficiency of Rep. Prog. Phys. 80, 016503 (2016). Electron-Polariton Scattering, Phys. Rev. B 72, 075340 [8] N. Engheta, Circuits with Light at Nanoscales: Optical (2005). Nanocircuits Inspired by Metamaterials, Science 317, 1698 [26] T. Brunhes, R. Andr´e, A. Arnoult, J. Cibert, and A. Wasiela, (2007). Oscillator Strength Transfer from X to X þ in a CdTe [9] H.-T. Lim, E. Togan, M. Kroner, J. Miguel-Sanchez, and A. Quantum-Well Microcavity, Phys. Rev. B 60, 11568 (1999). Imamoğlu, Electrically Tunable Artificial Gauge Potential [27] R. Rapaport, R. Harel, E. Cohen, A. Ron, E. Linder, and for Polaritons, Nat. Commun. 8, 1 (2017). L. N. Pfeiffer, Negatively Charged Quantum Well Polar- [10] T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, itons in a GaAs/AlAs Microcavity: An Analog of in a Topological Polaritons, Phys. Rev. X 5, 031001 (2015). Cavity, Phys. Rev. Lett. 84, 1607 (2000). [11] O. Bleu, D. D. Solnyshkov, and G. Malpuech, Measuring [28] R. Rapaport, E. Cohen, A. Ron, E. Linder, and L. N. the Quantum Geometric Tensor in Two-Dimensional Pho- Pfeiffer, Negatively Charged Polaritons in a Semiconductor tonic and Exciton-Polariton Systems, Phys. Rev. B 97, Microcavity, Phys. Rev. B 63, 235310 (2001). 195422 (2018). [29] R. A. Suris, in Optical Properties of 2D Systems with [12] A. Guti´errez-Rubio, L. Chirolli, L. Martín-Moreno, F. J. Interacting Electrons, NATO Science Series, edited by García-Vidal, and F. Guinea, Polariton Anomalous Hall W. J. Ossau and R. Suris (Springer, Dordrecht, 2003) Effect in Transition-Metal Dichalcogenides, Phys. Rev. pp. 111–124. Lett. 121, 137402 (2018). [30] M. Combescot and J. Tribollet, Trion Oscillator Strength, [13] S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, Solid State Commun. 128, 273 (2003). R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, [31] M. Combescot, J. Tribollet, G. Karczewski, F. Bernardot, C. T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, Testelin, and M. Chamarro, Many-Body Origin of the Trion

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Line in Doped Quantum Wells, Europhys. Lett. 71, 431 and High Currents, J. Res. Natl. Inst. Stand. Technol. 100, (2005). 529 (1995). [32] M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. [44] S. Smolka, W. Wuester, F. Haupt, S. Faelt, W. Wegscheider, Kroner, E. Demler, and A. Imamoglu, Fermi Polaron- and A. Imamoglu, Cavity Quantum Electrodynamics with Polaritons in Charge-Tunable Atomically Thin Semicon- Many-Body States of a Two-Dimensional Electron Gas, ductors, Nat. Phys. 13, 255 (2017). Science 346, 332 (2014). [33] D. K. Efimkin and A. H. MacDonald, Many-Body Theory of [45] K. V. Kavokin, I. A. Shelykh, A. V. Kavokin, G. Malpuech, Trion Absorption Features in Two-Dimensional Semicon- and P. Bigenwald, Quantum Theory of Spin Dynamics of ductors, Phys. Rev. B 95, 035417 (2017). Exciton-Polaritons in Microcavities, Phys. Rev. Lett. 92, [34] D. K. Efimkin and A. H. MacDonald, Exciton-Polarons in 017401 (2004). Doped Semiconductors in a Strong Magnetic Field, Phys. [46] M. Onoda, S. Murakami, and N. Nagaosa, Hall Effect of Rev. B 97, 235432 (2018). Light, Phys. Rev. Lett. 93, 083901 (2004). [35] O. Cotleţ, F. Pientka, R. Schmidt, G. Zarand, E. Demler, and [47] A. Kavokin, G. Malpuech, and M. Glazov, Optical Spin A. Imamoglu, Transport of Neutral Optical Excitations Hall Effect, Phys. Rev. Lett. 95, 136601 (2005). Using Electric Fields, Phys. Rev. X 9, 041019 (2019). [48] C. Leyder, M. Romanelli, J. P. Karr, E. Giacobino, T. C. H. [36] S. Ravets, P. Knüppel, S. Faelt, O. Cotlet, M. Kroner, W. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and Wegscheider, and A. Imamoglu, Polaron Polaritons in the A. Bramati, Observation of the Optical Spin Hall Effect, Integer and Fractional Quantum Hall Regimes, Phys. Rev. Nat. Phys. 3, 628 (2007). Lett. 120, 057401 (2018). [49] A. Amo, T. C. H. Liew, C. Adrados, E. Giacobino, A. V. [37] See Supplemental Material at http://link.aps.org/ Kavokin, and A. Bramati, Anisotropic Optical Spin Hall supplemental/10.1103/PhysRevX.10.011040 for additional Effect in Semiconductor Microcavities, Phys. Rev. B 80, information about the optical setup, electrical and polari- 165325 (2009). zation properties of the sample, the theoretical model, and [50] M. Maragkou, C. E. Richards, T. Ostatnický, A. J. D. spin density gradients, which includes Refs. [38–41]. Grundy, J. Zajac, M. Hugues, W. Langbein, and P. G. [38] D. Fritzsche, Heterostructures in MODFETs, Solid-State Lagoudakis, Optical Analogue of the Spin Hall Effect in Electron. 30, 1183 (1987). a Photonic Cavity, Opt. Lett. 36, 1095 (2011). [39] M. J. Kane, N. Apsley, D. A. Anderson, L. L. Taylor, [51] K. Lekenta, M. Król, R. Mirek, K. Łempicka, D. Stephan, and T. Kerr, Parallel Conduction in GaAs=AlxGa1−xAs R. Mazur, P. Morawiak, P. Kula, W. Piecek, P. G. Modulation Doped Heterojunctions, J. Phys. C 18, 5629 Lagoudakis, B. Piętka, and J. Szczytko, Tunable Optical (1985). Spin Hall Effect in a Liquid Crystal Microcavity, Light Sci. [40] M. Reed, W. Kirk, and P. Kobiela, Investigation of Parallel Appl. 7, 1 (2018). Conduction in GaAs=AlxGa1−xAs Modulation-Doped [52] A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. Structures in the Quantum Limit, IEEE J. Quantum Elec- De Giorgi, D. Ballarini, K. West, L. N. Pfeiffer, D. D. tron. 22, 1753 (1986). Solnyshkov, D. Sanvitto, and G. Malpuech, Direct Meas- [41] M. Steger, G. Liu, B. Nelsen, C. Gautham, D. W. Snoke, urement of the Quantum Geometric Tensor in a Two- R. Balili, L. Pfeiffer, and K. West, Long-Range Ballistic Dimensional Continuous Medium, arXiv:1901.03219. Motion and Coherent Flow of Long-Lifetime Polaritons, [53] S. M. Girvin, Spin and Isospin: Exotic Order in Quantum Phys. Rev. B 88, 235314 (2013). Hall Ferromagnets,inThe Multifaceted , edited [42] P. F. Fontein, J. A. Kleinen, P. Hendriks, F. A. P. Blom, J. H. by M. Rho and I. Zahed (World Scientific, Singapore, Wolter, H. G. M. Lochs, F. A. J. M. Driessen, L. J. Giling, 2015), pp. 351–365. and C. W. J. Beenakker, Spatial Potential Distribution in [54] P. Knüppel, S. Ravets, M. Kroner, S. Fält, W. Wegscheider, GaAs=AlxGa1−xAs Heterostructures under Quantum Hall and A. Imamoglu, Nonlinear Optics in the Fractional Conditions Studied with the Linear Electro-optic Effect, Quantum Hall Regime, Nature (London) 572,91 Phys. Rev. B 43, 12090 (1991). (2019). [43] M. Cage and C. Lavine, Potential and Current Distributions [55] P. Knüppel, ETH Zurich Research Collection, http://hdl Calculated Across A Quantum Hall-Effect Sample at Low .handle.net/20.500.11850/387816 (2020).

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